6.4. ENERGY LEVELS OF SUBATOMIC PARTICLES 365
with
(6.4.26) Eˆ=i ̄h
∂
∂t, pˆ=i ̄h(~σ·∇).As consider the massless bound states in weak and strong interactions, the Hermitian
operators in (6.4.26) are replaced by
(6.4.27) Eˆ=i ̄h
∂
∂t−gA 0 , pˆ=i ̄h(~σ·~D),where~D= (D 1 ,D 2 ,D 3 )is as in (6.4.22).
In Section6.4.1, we knew that the mediators such as photons and gluons consist of two
massless weaktons, which are bound in a small ballBrby the weak and strong interactions.
Hence the Weyl spinorψof each weakton is restricted in a small ballBr, i.e.
ψ= 0 ∀x6∈Br,which implies the boundary condition
(6.4.28) ψ|∂Br= 0.
However, in mathematics the boundary problem for the Weyl equations generated by
(6.4.25) and (6.4.27) given by
(6.4.29)
i ̄h∂ ψ∂t=ihc ̄ (~σ·~D)ψ+gA 0 ψ,
ψ|∂Br= 0 ,is in general not well-posed, i.e (6.4.29) has no solution for a given initial valueψ( 0 ) =ψ 0
in general. Hence, for a massless fermion system with the boundary condition (6.4.28), we
have to consider the relation
(6.4.30) pE=cp^2 ,
which is of first order in timet. It is known that the operator ˆpEˆis Hermitian if and only if
pˆEˆ=Eˆpˆ.Note thatEˆ=i ̄h∂/∂t−gA 0 , and in general
pAˆ 06 =A 0 pˆ.Hence, in order to ensure ˆpEˆbeing Hermitian, we replace ˆpA 0 by^12 (pAˆ 0 +A 0 pˆ), i.e. take
(6.4.31) pˆEˆ=ih ̄pˆ
∂
∂t−
g
2(pAˆ 0 +A 0 pˆ), pˆ=ihc ̄ (~σ·~D).Then by Postulate6.5, from (6.4.30) and (6.4.31) we derive the boundary problem of
massless system in the following form
(6.4.32)
(~σ·~D)∂ ψ∂t =c(~σ·~D)^2 ψ−ig 2 h ̄{(~σ·~D),A 0 }ψ,
ψ|∂Ω= 0 ,